On most current monitors there is a source button that allows you to switch between different outputs like VGA, DVI and HDMI. So for example you could have your PC hooked up DVI, Xbox VGA, and PS3 HDMI. They you can switch between them.

Actually you can I believe. I don't have an Xbox, but I am assuming you have component outputs. You can use a component to VGA cable that will allow you to use a VGA monitor on a component source. Once you convert to VGA that way, you can go through a KVM. A manual switch KVM will work and there are some automated ones that you can use your keyboard to flip between by hitting the scroll lock or some other hotkey.

You can also install a capture card on your PC.
I currently use one that has an HDMI and a Component/Composite input. I have my PS3 plugged into the Component input (Not XBox, but it's still applicable, just do the same thing). The card comes with software that you just open and it displays on your monitor whatever's being fed into the card. Can alt/tab to your desktop with your XBox still running.